Let $l$ and $w$ be the length and width of picture respectively that is being enclosed within the picture

Length and width of frame is given as $18$ and $15$ respectively.

length and width of the picture have the same ratio as the length and width of the frame, which means $\frac{l}{w} = \frac{18}{15} = \frac{6}{5} \Rightarrow w = \frac{5}{6}l$

Area of the frame itself = Total area of entire square - Area of the picture = $18*15 - l*b$

The frame encloses a rectangular picture that has the same area as the frame itself, meaning

$18*15 - lb = lb$

$18*15 = 2lb = 2(l)(\frac{5}{6}l) = \frac{5}{3}l^2$

$l^2 = \frac{18*15*3}{5} = 18*3*3 = 9*2*9 = 9^2*2$

$l = \sqrt{9^2*2} = 9\sqrt{2}$

Hence, Answer is A